Linearization of holomorphic vector fields and a characterization of cone singularities
A local homogeneous cone is a triple (X,0,\(g)\) consisting of a germ of a complex space (X,0) and a finite dimensional Lie algebra \(g\) of holomorphic vector fields on (X,0) satisfying the following conditions: i) \(X\setminus \{0\}\) is smooth ii) \(\zeta (0)=0\) for all \(\zeta\) \(\in g\) and iii) the complex tangent bundle of \(X\setminus \{0\}\) is generated by \(g.\) The main result of the paper is the characterization of the germs (X,0) and the Lie algebras \(g\) with the property that (X,0,\(g)\) is a local homogeneous cone. A local homogeneous cone is shown to be linearizable, i.e. it is equivariantly isomorphic to the germ of the vertex of a homogeneous cone over a projective rational manifold. For the case where (X,0) is smooth, this means that the Lie algebra \(g\) consists of linear vectorfields and homogeneous vectorfields of a fixed degree. The main tool of the proof is the \(g\)-anticanonical map. In particular a special case of the Lipman-Zariski problem in dimension two is settled.
linearization of holomorphic vector fields
scientific article; zbMATH DE number 4044209
scientific article